Hi illuminati5288,

The word "distinct" in the prompt, and in Fact 1, means DIFFERENT.

When combining the two Facts, there are only 5 distinct points in the plane. Even if there were "duplicate points" (e.g.. on a co-ordinate plane, point A is at (1,1) and point B is also at (1,1)), triangle ACD and triangle BCD would be the SAME triangle, so you would not count it twice.

Since no group of three points are collinear, that means we can select ANY group of 3 (of the 5 points) to make a triangle. Thus, there are 5!/3!2! = 10 possible distinct triangles that can be formed.

If you're having trouble with the conceptual ideas behind this prompt, then you could try drawing 5 random points on a piece of paper. As long as you don't have 3 of them on the same line, then you'll end up with 10 distinct triangles.

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

illuminati5288 wrote:
S is set of points in a plane. How many distinct triangles can be drawn that have three of the points in S as vertices?

(1) The number of distinct points in S is 5.
(2) No three of the points in S are collinear.

We are given that S is a set of points in the plane and we must determine how many distinct triangles can be drawn with three of the points in S as vertices. So essentially, we must determine how many distinct triangles can be drawn with the points provided.

Statement One Alone:

The number of distinct points in S is 5.

Using the information in statement one, it may be tempting to conclude that the number of triangles that can be drawn is 5C3 = (5 x 4 x 3)/3! = 10 triangles. However, because we do not know the positioning of the points, we cannot actually say that 10 distinct triangles can be created. Let's say, for instance, that all the points were collinear, which means that they are all located on one line. If that were the case, we would not be able to create any triangles. Thus, statement one is not sufficient to answer the question. We can eliminate answer choices A and D.

Note: We were able to determine that 10 triangles could be formed with 5 points if and only if no 3 points are collinear. Only then would the number of triangles be 5C3 = 10.

Statement Two Alone:

No three of the points in S are collinear.

Using the information in statement two, we cannot answer the question because we do not know how many points are in S. We can eliminate answer choice B.

Statements One and Two Together:

Using the information from statements one and two we know that we have 5 points in the plane and that no three points are collinear. Thus, we can determine that the number of triangles that can be created in the plane is

5C3 = 10.

Answer: C